Carl Paladino’s equally fiery campaign manager twice offered to quit following the anti-gay stump speech that left even the candidate’s family offended — but the Republican nominee for governor refused to take the resignation.

Michael Caputo tried to quit Thursday — less than a week after the controversial speech drew widespread criticism from both sides of the political aisle.

Immediately after the speech that sent shockwaves from Greenwich Village to San Francisco, Caputo admitted that he helped write the address, which included a reference to kids being “brainwashed” into thinking homosexuality is OK.

Another draft of the speech — which Paladino didn’t read but was handed to the media — referred to “dysfunctional homosexuals.”

Those anti-gay rants sparked a political firestorm and threw Paladino’s campaign into a tailspin.

After initially refusing to back down, Paladino apologized — but only after saying his Democratic rival Andrew Cuomo shouldn’t have taken his kids to a “disgusting” gay-pride parade.

Meanwhile, Cuomo — with a newly energized liberal base seething at the thought of Paladino in the governor’s mansion — released a new ad claiming to be a unifying figure who will clean up Albany.

“I know we can turn this state around. We’ll have to be strong, but we’ll also have to be smart and work together. That’s a New York we can be proud of,” he said.